PHYSICAL REVIEW D
VOLUME 56, NUMBER 8
15 OCTOBER 1997
Cosmic strings in an open universe: Quantitative evolution and observational consequences
P. P. Avelino*
Centro de Astrofı́sica, Universidade do Porto, Rua do Campo Alegre 823, PT-4150 Porto, Portugal
R. R. Caldwell†
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104
C. J. A. P. Martins‡
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street,
Cambridge CB3 9EW, United Kingdom
~Received 19 February 1997!
The cosmic string scenario in an open universe is developed—including the equations of motion, a model of
network evolution, the large angular scale cosmic microwave background ~CMB! anisotropy, and the power
spectrum of density fluctuations produced by cosmic strings with dark matter. We first derive the equations of
motion for a cosmic string in an open Friedmann-Robertson-Walker ~FRW! space-time. With these equations
and the cosmic string stress-energy conservation law, we construct a quantitative model of the evolution of the
gross features of a cosmic string network in a dust-dominated, V,1 FRW space-time. Second, we apply this
model of network evolution to the results of a numerical simulation of cosmic strings in a dust-dominated,
V51 FRW space-time, in order to estimate the rms temperature anisotropy induced by cosmic strings in the
CMB. By comparing to the COBE-DMR observations, we obtain the normalization for the cosmic string mass
per unit length m as a function of V. Third, we consider the effects of the network evolution and normalization
in an open universe on the large scale structure formation scenarios with either cold or hot dark matter ~CDM,
HDM!. The string1HDM scenario for V,1 appears to produce too little power on scales k*1 Vh 2 /Mpc. In
a low density universe the string1CDM scenario is a better model for structure formation. We find that for
cosmological parameters G5Vh;0.1–0.2 in an open universe the string1CDM power spectrum fits the shape
of the linear power spectrum inferred from various galaxy surveys. For V;0.2–0.4, the model requires a bias
b*2 in the variance of the mass fluctuation on scales 8 h 21 Mpc. In the presence of a cosmological constant,
the spatially flat string1CDM power spectrum requires a slightly lower bias than for an open universe of the
same matter density. @S0556-2821~97!05920-1#
PACS number~s!: 98.80.Cq, 11.27.1d, 98.65.Dx, 98.70.Vc
I. INTRODUCTION
Electronic address: caldwell @ dept.physics.upenn.edu
Also at C.A.U.P., Rua do Campo Alegre 823, 4150 Porto, Portugal. Electronic address: [email protected]
power spectrum, while producing too much small scale
power in the case V51 @9#, appears to fit the the shape of
the power spectrum estimated from the various threedimensional galaxy redshift surveys for V,1 @5,6#. Hence,
we aim to develop the tools necessary to study cosmic
strings in an open universe.
The outline of this paper is as follows. In Sec. II we
construct a background cosmology composed of a dustdominated, V,1 FRW space-time. We derive the cosmic
string equations of motion and the energy conservation equation in an open universe, and discuss the effects of the spatial
curvature and rapid, curvature-dominated expansion on the
density of strings through a simple solution of these equations.
We next construct an analytic model of the long string
evolution in an open universe. While not as sophisticated as
the model developed by Austin, Copeland, and Kibble @10#,
we improve on earlier work @11# by following the procedure
of Martins and Shellard ~MS! @12#, which treats the mean
string velocity as a dynamical variable. We should point out
that the MS model provides an accurate description of the
behavior of a cosmic string network seen in numerical simulations of radiation- through matter-dominated expansion in
the case V51. Since no such simulations exist in the case of
an open universe, the model presented in this paper is an
0556-2821/97/56~8!/4568~10!/$10.00
4568
Cosmic strings are topological defects which may have
formed in the very early universe and may be responsible for
the formation of large scale structure observed in the universe today @1–3#. In order to test the hypothesis that the
inhomogeneities in our universe were induced by cosmic
strings one must compare observations of our universe with
the predictions of the cosmic string model. To date, most
work on the cosmic string scenario has been carried out with
a background cosmological model which is a spatially flat,
V51 Friedmann-Robertson-Walker ~FRW! space-time. ~See
@4–7# for other work on open scenarios with defects.!
Observational evidence indicates that to within 95% confidence, the present-day cosmological density parameter lies
in the range 0.2,V,2, and is most likely less than or equal
to unity @8#. This reason alone is enough motivation to investigate open cosmologies. We are further compelled when
we recognize that the string1cold dark matter ~CDM! linear
*Electronic address: pedro @ pulsar.astro.up.pt
†
‡
56
© 1997 The American Physical Society
56
COSMIC STRINGS IN AN OPEN UNIVERSE: . . .
extrapolation. Nevertheless, the onset of curvature domination in the present case is qualitatively similar to the
radiation-matter transition; in both cases, the dominant dynamical effect during the transition is due to the shift in the
time dependence of the scale factor. Hence, we expect that
the model developed in this section, which includes the effects of curvature-driven expansion and spatial curvature on
the the string equations of motion, will be sufficient to provide a good description of cosmic string evolution in an open
universe—although future numerical work will undoubtedly
be required to test this model.
In Sec. III, by numerically solving the evolution equations, we find that with the onset of curvature domination,
the mean string velocity and energy density decay rapidly.
We also note that there does not appear to be a scaling solution for the gross features of the string network, as occurs
in a spatially flat, V51 cosmology. Similar work has also
been carried out recently by Martins @13#. In Sec. IV we
construct a semianalytic model of the CMB anisotropy induced by strings in an open cosmology. We obtain the normalization of the string mass per unit length m , as a function
of V, by comparing with the Cosmic Background Explorer
~COBE! Differential MIcrowave Radiometer ~DMR! observations. Next, we consider the effect of the new normalization on the large scale structure power spectrum when V,1
by adapting the Albrecht-Stebbins @9# semianalytic model for
the string1CDM and hot dark matter ~HDM! scenarios.
While the power spectrum does not completely specify the
non-Gaussian fluctuation patterns generated by cosmic string
wakes, it serves as a useful gauge of the viability of the
scenario. We find that in an open universe, the string1HDM
spectrum suffers from a lack of power on small scales, compared to the linear power spectrum estimated from various
galaxy redshift surveys. However, the string1CDM spectrum in an open universe with bias b*2 appears to fit the
observed power spectrum. Finally, in Sec. VI, we consider
the case of a spatially flat, low matter density universe with a
cosmological constant. Applying the same tools that we have
developed for the study of an open universe, we obtain the
CMB normalization of the mass per unit length. We find that
the string1CDM power spectrum requires a slightly lower
bias than for an open universe with the same matter density.
We conclude in Sec. VII.
Throughout this paper we have set the speed of light to
unity, c51, and adopted the convention H 0 [100h
km sec21 Mpc21 .
R5a ~ t ! R c 5H 21 u 12V u 21/2.
F
g ab 5g m n x m ,a x n ,b → g tt 5a 2 ~ t ! 12
g ss 52a 2 ~ t !
2
2
)/R 2c ,
~ 11Kr 2 ! 2
~ 11Kr 2 ! 2
G
,
~2.3!
.
E
m
E
r5
m
a2
d s A2 gg ab x t ,a x t ,b d 3 @ x2x~ t , s !#
5
a2
ds
5
m
a2
E
x8
A~ 11Kr 2 ! 2 2ẋ2
d 3 @ x2x~ t , s !#
d s ẽ d 3 @ x2x~ t , s !# ⇒ ẽ [
x8
A~ 11Kr 2 ! 2 2ẋ2
.
~2.4!
Here, ẽ is the comoving coordinate length of string per unit
s . The integral above tells us that the total energy in string is
not simply the total length of string, weighted by a relativistic g factor, times the mass per unit length m . The spatial
curvature also has an effect, as there is a contribution to the
energy by the Kr 2 term for strings with large spatial extent.
This coupling of curvature to the string energy will have an
important effect on the evolution of very large strings.
The equations of motion for cosmic string in the spacetime, Eq. ~2.1!, are given by ~generalizing Eq. 6.1.12 of @2#!
x m,a ;a 1G mnr g ab x n,a x r,b 50,
~2.1!
where r 5(x 1y 1z
K521 for an open FRW
space-time, and the coordinates lie in the range $ x,y,z %
P(2R c ,R c ). Note that for K521,0,11 the spatial sections
are H 3 , R 3 , S 3 , respectively. The radius of spatial curvature
is given by
2
x8 2
ẋ2
The indices a,b denote coordinates t , s on the string world
sheet, with •5 ] t and 8 5 ] s , where s is a parameter along
the string. We have chosen the gauge such that the time
parameter along the string coincides with the conformal time
t , and ẋ•x8 50.
Referring to @2# ~Eq. 6.1.15! for the string stress-energy
tensor, we find that the energy density in string is given by
To begin our investigation of cosmic strings in an open
FRW space-time, we must construct a background cosmology. We find it convenient to use the metric
2
~2.2!
Thus, a universe with a density close to critical has a very
large radius of curvature and is very flat, whereas in a low
density universe the curvature radius is comparable to the
Hubble radius. The cosmological time t is related to the conformal time by t5 * d t a( t ).
In this space-time, the induced metric on the cosmic string
world sheet is
II. OPEN UNIVERSE
ds 2 5a 2 ~ t !@ d t 2 2 ~ 11Kr 2 ! 22 ~ dx 2 1dy 2 1dz 2 !# ,
4569
ẋ2
ȧ
˙
m 5 t → ẽ 522 ẽ
,
a ~ 11Kr 2 ! 2
S
~2.5!
D S D
ȧ
ẋ2
1 x 8i 8
m 5i→ẍ i 12 ẋ i 12
2
a
ẽ ẽ
~ 11Kr 2 ! 2
5
2Ka 2 H 2 u 12V u i 2
@ x ~ ẋ 2 ẽ 22 x8 2 ! 22ẋ i ~ x•ẋ!
11Kr 2
12 ẽ 22 x 8 i ~ x•x8 !# .
~2.6!
4570
56
P. P. AVELINO, R. R. CALDWELL, AND C. J. A. P. MARTINS
ing guarantees that the denominator of ẽ , (12r 2 ) 2 2ẋ 2 , is
always positive; for large r, ẋ must decrease. In short, the
curvature-driven expansion serves to damp the string motion.
We will move on to concentrate on Eq. ~2.5!, the conservation equation, for the present. Making the definition
v 2r 5
m
a2
E
d s ẽ ẋ2 d 3 @ x2x~ t , s !# ,
~2.8!
for the mean-squared string velocity, we obtain the energy
density conservation equation
ȧ m
ȧ
ṙ 12 r 522
a
a a2
FIG. 1. The evolution of a circular cosmic string loop formed at
t5t eq with an initial radius R loop510t eq , in a universe with V51
and 0.2 given by the solid and dotted lines, respectively. The top
panel shows the evolution of the radius in units of t eq versus
log10(a/a eq). The bottom two panels show the evolution of the velocity. An expanded scale shows the first oscillations as the loop
enters the horizon, after which we show only the maximum velocity
in each period of oscillation.
The above equations represent the first main result of this
paper; by setting K521, Eqs. ~2.5! and ~2.6! give the equations of motion for cosmic strings in an open FRW spacetime. Similarily, K511, 0 give the equations of motion in
closed and flat FRW space-times. Hereafter we will only
consider the case of an open universe.
We can get a better idea of the effects of the rapid expansion and space-time curvature by studying a simple solution
of these microscopic equations. For our purposes it is sufficient to consider dust-dominated expansion only, for which
the scale factor is given by
a~ t !5
V sinh2 ~ t /2!
H 0 ~ 12V ! 3/2
.
~2.7!
Here H 0 and V are the present-day Hubble constant and
cosmological density parameter. We have solved Eq. ~2.6!
for the case of a circular loop with an initial physical radius
R loop(t eq)510t eq and velocity v (t eq)50 at the time of
radiation-matter equality. The evolution of the loop radius
and velocity, for the cases V51, 0.2, is shown in Fig. 1. At
early times the evolution is indistinguishable, as the loop is
conformally stretched by the expansion and picks up speed.
As the loop falls inside the Hubble horizon, it begins to
oscillate. We see that the frequency of oscillation, while constant relative to H, is higher relative to t 21 in an open universe owing to the faster expansion rate. This is crucial because oscillating loops in an expanding universe lose a
constant fraction of their energy in each period @12,14#.
Hence, the average velocity at late times is smaller than the
spatially flat value v 2flat51/2. Incidentally, the velocity damp-
E
d s ẽ
ẋ2
~ 12r 2 ! 2
d 3 @ x2x~ t , s !# .
~2.9!
For r→1 (0,r 2 ,1) with fixed ẋ2 the term on the righthand side ~RHS! of Eq. ~2.9! becomes large and negative.
Hence, this confirms that the effect of the spatial curvature
will be to enhance the dilution of the energy density in long
strings due to the expansion.
Let us examine Eq. ~2.9! more closely. The integrand on
the RHS is just the coordinate energy ẽ times the velocity
squared, weighted by a factor of the ratio r of the string
length to the curvature scale. By averaging over strings of
length scale l, we may rewrite Eq. ~2.9! as
ȧ
ȧ
ṙ l 12 r l 522 r l ^ v 2 & @ 12 ~ l/R ! 2 # 22 .
a
a
~2.10!
The ratio l/R determines the importance of the curvature
contribution. For strings much smaller than the curvature
scale, l!R, we obtain the usual flat space energy density
conservation equation. For very large strings, l→R. Hence,
string with support on very large scales samples more of, or
is more tightly coupled to, the curvature. Crudely, the effect
is that the long string energy density is dissipated more rapidly as the space-time expands.
III. QUANTITATIVE STRING EVOLUTION
We now build on the work of Martins and Shellard @12# to
construct an analytic model of long string network evolution
in an open universe. As carried out by MS, we treat the
average string velocity as well as the characteristic string
length scale as dynamical variables. For long strings, the
characteristic length scale is related to the network density of
long strings by r L 5 m L 22 . Hence, we obtain, from Eq.
~2.10!,
dL
1
5LH $ 11 v 2 @ 12 ~ 12V !~ LH ! 2 # 22 % 1 c̃ v . ~3.1!
dt
2
The phenomenological loop chopping efficiency parameter c̃
models the transfer of energy from the long strings to loops.
Next, an evolution equation for the velocity may be obtained
by differentiating Eq. ~2.8! and using Eq. ~2.6!:
56
COSMIC STRINGS IN AN OPEN UNIVERSE: . . .
4571
dv
k
5 $ @ 12 ~ 12V !~ LH ! 2 # 2 2 v 2 %
dt
L
22H v $ 12 v 2 @ 12 ~ 12V !~ LH ! 2 # 22 % .
~3.2!
As in MS @cf. Eqs. ~2.40! and ~2.41! of @12## the parameter k
~MS use k) has been introduced to describe the presence of
small scale structure on the long strings. Equations ~3.1! and
~3.2! are the second main result of this paper. Again, in the
limit V→1, the flat space evolution equations @Eqs. ~2.20!
and ~2.38! of @12## are obtained. ~Note that the above equations are equally valid in a closed, V.1 FRW space-time.!
We have omitted the friction damping terms due to the interaction of the cosmic strings with the hot cosmological
fluid, which are important only near the time the strings were
formed. Similar equations are given in Ref. @13#, although it
was assumed that the contribution of the curvature terms is
negligible.
In a spatially flat, V51 FRW space-time, scaling solutions may be found for which v̇ 50 and L/t5const. In a
dust-dominated era, the Allen-Shellard numerical simulation
suggests the values k 50.43 and c̃ 50.15 ~note that MS used
k 50.49 and c̃ 50.17—these values give the same mean velocity, but the string density is closer to the Bennett-Bouchet
value, though is consistent within the quoted error bars!.
In an open, V,1 FRW space-time, for normal types of
matter ~i.e., dust or radiation!, H does not decay like t 21 .
Ignoring the curvature terms, the solution of Eq. ~3.2! for
which v 5const is inconsistent with L}t from Eq. ~3.2!, and
inconsistent with L}H 21 from Eq. ~3.1!. As noted in Ref.
@13#, it does not appear possible to find a scaling solution in
an open universe.
We have numerically solved the evolution equations ~3.1!
and ~3.2! with the expansion scale factor given by Eq. ~2.7!.
We choose the initial conditions for L/t and v to be given by
the V51 dust era scaling solution of v 50.61 and L/t50.53.
Our reasoning is that at early times, when the scale factor
behaves as a}t 2/3, the evolution is indistinguishable from an
V51 space-time. Only at late times is the effect of the
curvature-dominated expansion important. Similarily, we assume that the coefficients k and c̃ , describing the small scale
structure and chopping efficiency, are unchanged from their
dust-era values. This is somewhat unrealistic, since these parameters differ even between the radiation and dust eras. Up
to radiation-matter equality, however, work by MS has
shown that the effect on the evolution is dominated by the
change in the expansion rate rather than the shift in the parameters. Hence, we expect our model to be reliable for the
observationally allowed values of V as we do not follow the
evolution too far beyond matter-curvature equality.
Sample results are displayed in Fig. 2. We see that the
rms velocity and string density decrease rapidly at late times
when the curvature begins to dominate the expansion. For
V50.2, the velocity drops to v 50.41, and the length scale
grows to L/t50.63. By letting V→1 in Eqs. ~3.1! and ~3.2!,
we find final values v 50.44 and L/t50.64. Hence, the rapid,
curvature-dominated expansion is the main cause of the departure from scaling, since the spatial curvature terms contribute only a &10% effect to the evolution. Consequently,
our results depend only very weakly on the Hubble param-
FIG. 2. The evolution of the average string velocity and the
characteristic length scale of long strings in an open FRW spacetime with V51.0, 0.6, and 0.2 given by the solid, dashed, and
dotted curves, respectively. The horizontal axis is the logarithm of
the cosmological time t. As the expansion becomes curvature dominated, the average velocity decays and the characteristic string
length scale grows. As a result, the number of long strings in a box
of linear dimension t decreases, although the string energy density
relative to the background energy density grows.
eter h. As pointed out by Martins, the energy density in long
strings is actually growing relative to the background cosmological fluid. At sufficiently late times, the strings will come
to dominate the energy density of the universe. This deviation from the scaling solutions should have an important effect on the large angle CMB anisotropy due to cosmic
strings.
IV. CMB ANISOTROPY
While it is beyond our means to simulate the evolution of
cosmic strings in an V,1, dust-dominated FRW space-time
at present, we may nevertheless adapt our model for the
quantitative evolution of a string network to estimate the
amplitude of CMB temperature anisotropy induced by cosmic strings.
We would like to determine the COBE-smoothed rms
temperature anisotropy due to cosmic strings in an V,1
cosmology. Hence, we must compute C(0°,10°), the 0° angular separation correlation function smoothed over 10° in
the manner of COBE. To do so, we will make the following
simplifying assumptions
~1! The large angle CMB anisotropy is due to the gravitational perturbations caused by cosmic strings along the line
of sight out to the surface of last scattering.
~2! The mean, observer-averaged angular correlation
function may be written as the sum of the contributions by
strings located in the time interval @ t,t1 d t # ; the contribution
due to strings separated by an interval larger than the characteristic time scale, d t*L, is negligible.
~3! The effect of the negative spatial curvature in the open
P. P. AVELINO, R. R. CALDWELL, AND C. J. A. P. MARTINS
4572
56
interval @ z,z1 d z # , we must restrict our use to a time resolution d t*L, greater than the characteristic time scale.
The negatively curved spatial sections of the open FRW
space-time lead to a generic suppression of large-angle correlations. ~See Ref. @18# and references therein for more discussion.! We may understand this effect by considering that
an object with angular size u 21 at a redshift z from an observer in an open FRW space-time subtends a smaller angle
u 21 , u 0 than the angle of the same object from the same
redshift in a spatially flat FRW space-time. ~The subscripts 0,
21 refer to the sign of the spatial curvature.! We may express this relationship between the angles subtended as
u 21 [ f ~ u 0 ,z,V !
F
G
V 2 ~ 11z2 A11z !
u0
52arcsin sin
.
2 Vz1 ~ 22V !~ 12 A11Vz !
~4.2!
FIG. 3. The CMB-normalized power spectrum P(k) of density
fluctuations produced by cosmic strings with HDM ~left! and CDM
~right! are presented for V51.0, 0.4, and 0.2, given by the thick
solid, long-dashed, and short dashed curves, respectively. For all
cases, we have used h50.7. In the top panels, the thin solid line is
the standard CDM spectrum normalized to COBE following @34#.
In the lower panels, the data points are the PD reconstruction of the
linear power spectrum, with the amplitude rescaled }V 20.3. In the
bottom two V,1 string1CDM panels, the thin solid line shows the
CMB-normalized power spectrum for the case of a cosmological
constant with the same matter density. A bias b;2 –4 is necessary
to obtain s 8 ;1. In the presence of a cosmological constant, a
smaller bias is required.
universe is to shift temperature anisotropy correlations to
smaller angular scales than in a spatially flat universe.
~4! The mean rms temperature anisotropy contributed in a
time interval d t is proportional to the density of strings
present and the mean string velocity during that interval.
This is similar to Perivolaropoulos’ model @15# in which the
CMB anisotropy is a superposition of random impulses due
to the Kaiser-Stebbins effect @16#, for which d T}8 p G m v ,
for each long string present.
Given the first two assumptions, the correlation function
may be written as
C ~ u ,z ls ! 5
E
z ls
0
dz C ~ u ! ,z .
~4.1!
Here, C( u ,z ls ) is the temperature correlation function contributed by strings out to the redshift of last scattering, z ls .
The function C( u ,z) has been tabulated from the numerical
simulation of CMB anisotropy induced by cosmic strings in
an V51, dust-dominated FRW space-time ~see Fig. 3 of
@17# where we observe that the dominant contribution to the
rms anisotropy for V51 occurs within a redshift z&10).
The function C( u ) ,z is obtained empirically by differentiating C( u ,z). This procedure does not rely on assumption ~2!
above. However, in order to interpret C( u ) ,z as the contribution to the angular power spectrum due to strings in the
As a result, u 21 < u 0 for all z>0 and V<1. In the limit
V→1 or z→0, Eq. ~4.2! reduces to the identity, with
u 0 5 u 21 . In order to include the effect of the geometry on
the temperature anisotropy correlation function in an open
universe, we write
dC l
~ V ! 52 p
dz
E
p
0
d ~ cosu 21 ! P l ~ cosu 21 ! C ~ u 0 ! ,z .
~4.3!
By shifting the argument of the Legendre polynomial to
smaller angles, correlations on a particular angular scale are
associated with a larger l mode in an open than in a flat
FRW space-time.
To implement our final assumption above, we model the
effect of the curvature-dominated expansion on the correlation function by weighting the contribution at different redshifts using our model of quantitative string evolution:
Cl ~ V !5
E F
z ls
0
dz
G
r L ~ V,z !
v 2 ~ V,z ! dC l
~ V !.
r L ~ V51,z ! v 2 ~ V51,z ! dz
~4.4!
Hence, the moments of the correlation function, which is
proportional to ( d T) 2 , are weighted by two powers of the
string velocity relative to the V51 value. We model the
contribution of the N long strings in each volume to the
temperature amplitude as AN, so that only one factor of the
string density relative to the V51 value is included above.
The functional dependence of r L and v on the redshift for a
given open cosmology is obtained by integrating Eqs. ~3.1!
and ~3.2!. Because these weights change on a time scale
comparable to or slower than L, assumption ~2! is satisfied.
We foresee that the CMB anisotropy will be diminished
due to the dilution of the string density, the decrease in mean
velocity, and the negative spatial curvature in an V,1 universe. The geometric effect due to the negative spatial curvature, in Eq. ~4.3!, will lead to a decrease in the amplitude
of the anisotropies generated at distances to the observer
which are large compared to the curvature length scale. The
dynamical effect due to the late time evolution of the string
network, in Eq. ~4.4!, will lead to a decrease in the amplitude
56
COSMIC STRINGS IN AN OPEN UNIVERSE: . . .
of the anisotropies generated at late times. The result is an
overall decrease in the amplitude of the CMB anisotropy
spectrum for a given mass per unit length.
We may estimate the normalization of the cosmic string
mass per unit length m in an open universe by comparing the
observations of COBE-DMR @19# with our predictions. We
carry out a procedure similar to that given in @17#, computing
the smoothed autocorrelation function
`
C ~ 0°,10°,V ! [
(
l 52
2 l 11
u G l u 2 u W l ~ 7° ! u 2 C l ~ V ! .
4p
~4.5!
Here, we smooth the temperature pattern first with the average DMR beam model window function G l ~tabulated values are given in @20#! which is approximately a 7° beam, and
second with a 7° full width at half maximum ~FWHM!
Gaussian window function W l (7°) for an effective smoothing of 10°. Thus, we find for the case V50.2,
26
G m 51.710.6
20.3310 . The effect of the spatial geometry on
the smoothed autocorrelation function is only ;20% for
V50.2; the dilution of the string density and the decrease in
the mean velocity due to the rapid expansion are the main
causes of the change in the rms anisotropy amplitude. We
have rescaled the error bars assessed in @17#, assigning no
errors due to the crudeness of our model. For V;1 this
seems reasonable; for low V we underestimate our uncertainty in the normalization. The empirical formula for the
CMB normalization of the string mass per unit length,
26 20.3
G m ~ V ! 5G m ~ V51 ! V 20.351.05 10.35
,
20.20310 V
~4.6!
fits our results to within 5% for 0.1<V<1. We stress that
our estimate of the normalization is valid, within the abovementioned error bars, insofar as Allen et al. @17# have accurately simulated the large angle CMB anisotropy induced by
realistic cosmic strings. This is the third main result of this
paper.
We take this opportunity to comment on the effect of an
open universe on the small angular scale CMB anisotropy
induced by cosmic strings. Although no firm predictions of
the high-l C l spectrum have been made to present, recent
work @21,22# has shed light on the qualitative features of the
spectrum. Based on numerical simulations, they observe a
feature near l ;100 attributed to the decay of vector perturbations smaller than the horizon scale on the surface of last
scattering. For higher l , there is a single, low, broad feature
~as opposed to the secondary oscillations predicted in inflationary scenarios! in the range l ;400–600, as conjectured
by Magueijo et al. @23#. In an open universe, the apparent
size of fluctuations near the surface of last scattering shift to
smaller angles as u }V 1/2. Hence, we expect the location of
the feature due to the decay of the vector perturbations to
shift as l ;100V21/2 towards smaller angular scales.
We end these comments on the small angular scale spectrum by adding that MS have shown that the transient in the
evolution of the long string density and velocity across the
radiation-matter transition, observed in the Bennett-Bouchet
numerical simulations @24,25#, may last as late as ;103 t eq
~see Figs. 18~c! and 18~d! of @12#!. In particular, the ratio
4573
r L / r crit , which is higher in the radiation era, does not settle
down to the matter era scaling value until ;103 t eq , and the
evolution of the mean velocity displays a peak near ;30t eq
before reaching the matter-era value. For low values of V
and h, the redshift of radiation-matter equality approaches
last scattering, so that this transient may have an important
effect on the small angle CMB anisotropy generated near the
surface of last scattering @26#.
V. LARGE SCALE STRUCTURE
Finally, we consider the large scale structure formation
scenario with cosmic strings. We will examine both the
HDM and CDM scenarios, by adapting the methods of Ref.
@9# to estimate the power spectrum of density fluctuations
produced by cosmic strings. While the effect of low V on
these string scenarios has been examined previously by Mahonen et al. @27# and Ferreira @6#, our contribution will be
the effect of the quantitative string evolution and the normalization of m on the power spectrum.
In the semianalytic model of Albrecht and Stebbins, the
power spectrum of density perturbations induced by cosmic
strings in an V51 universe is approximated by
P ~ k ! 516p 2 ~ 11z eq! 2 m 2
F~ k j /a ! 5
E
`
ti
u T ~ k; t 8 ! u 2 F~ k j /a ! d t 8 ,
2 2 x2
b S 2 @ 112 ~ k x /a ! 2 # 21 .
p2
j
~5.1!
In these equations, a is the scale factor which evolves
smoothly from radiation- to dust-dominated expansion, t i is
the conformal time at which the string network formed, and
T(k, t 8 ) is the transfer function for the evolution of the causally compensated perturbations @see Eq. ~2! of @9# and Eqs.
5.23 and 5.45 of @28##, specific to either CDM or HDM. In
the case of HDM, T(k, t 8 ) includes a term fit to numerical
calculations of the damping of perturbations by nonrelativistic neutrinos.
The parameters used in the Albrecht-Stebbins estimate of
the cosmic string power spectrum are given by
j [ ~ r L / m ! 21/2,
S[
b [ ^ v 2 & 1/2,
S
D
m 2r 2 m 2
mr
1
g bb b1
,
m
2 g b b b mm r
~5.2!
where x is the curvature scale of wakes, b b is the macroscopic bulk velocity of string, g b 5(12 b 2b ) 21/2, and m r is
the renormalized mass per unit length, which reflects the
accumulation of small scale structure on the string. The ‘‘I
model’’ developed by Albrecht and Stebbins uses the following values of the parameters:
era
radiation
dust
j /(a t )
x/j
b
bb
m r/ m
0.16
0.16
2.0
2.0
0.65
0.61
0.30
0.15
1.9
1.4
We note that the values of b , b b , and m r were taken from
the Allen-Shellard ~AS! simulation @29,30#. The values of j
4574
P. P. AVELINO, R. R. CALDWELL, AND C. J. A. P. MARTINS
and x , however, reflect an estimate based on the BennettBouchet ~BB! @24,25# and AS simulations. The radiation-era
values were used for the I model to determine the power
spectrum in a spatially flat, V51 universe.
While the I model closely resembles the AS and BB simulations, we might hope to make an improvement by including the effect of the evolution of the string network parameters across the transition from radiation- to dust-dominated
expansion @26#. As investigated by MS, the ratio L/t interpolates between the radiation- and dust-dominated scaling
values, whereas the mean velocity displays a short burst during which the string network rapidly sheds loops. @See Figs.
18~c! and ~d! of @12#.# Hence, we make the identifications
j 5L and x 52 j , and use the evolution of L/t to interpolate
between radiation- and dust-era values of m r and k , and use
b to guide b b . Note that the radiation-era values are
4pk3P~ k !5
11 ~ u 2 k ! 1 ~ u 3 k ! 2 1 ~ u 4 k ! 3 1 ~ u 8 k ! 4 1 ~ u 6 k ! u 7
u1
u2
u3
u4
u5
u6
u7
u8
HDM
CDM
6.8
6.8
4.7
4.7
4.4
4.4
1.55
1.55
2198
2198
2.46
0
6.6
0
3.2
0
where k is measured in units Vh 2 /Mpc and m 6 (V)
[G m (V)3106 obtained from Eq. ~4.6!. Numerical simulations of string seeded structure formation by Avelino @32#,
based on the Allen-Shellard simulation, find agreement with
Eq. ~5.3! in a flat universe on the limited range of scales
accessible to the simulation.
Sample power spectra for various cosmological parameters, constructed using Eq. ~5.3!, are shown in Fig. 3. The
string mass per unit length m in each of the curves has been
determined by the CMB normalization obtained from Eq.
~4.6! in Sec. IV. In the top panels, the power spectra for
h50.7 and V51.0, 0.4, and 0.2 are shown. For reference,
the standard CDM power spectrum @33# is also displayed. In
the three descending panels, the individual spectra are shown
with the Peacock-Dodds @34# ~PD! reconstruction of the linear power spectrum. For V,1 the reconstructed spectrum
has been scaled as }V 20.3 ~see Eq. 41 of @34#! for comparison.
We first consider structure formation by strings with
HDM. Based on the normalization of m obtained by @17#, we
see from Fig. 3 that the power spectrum approximately fits
the shape of the PD spectrum on large scales. As a gauge of
the string1HDM model for low V, we have computed the
variance of the excess mass fluctuation in a ball of radius
R58h 21 Mpc,
s 28 5
E
( c̃ , k )5(0.24,0.18) for the BB simulation and (0.22,0.16)
for the AS simulation. Applying the model of realistic network evolution to the power spectrum, we find that for the
same m , the only change is a ;30% boost in the power for
the AS values, which is consistent with the quoted uncertainties on the parameters measured in the simulations.
To adapt the power spectrum for an open universe, we
would like to use the transfer function T(k, t 8 ) appropriate
for V,1. In the present work, however, we will use the
V51 transfer function, which should be satisfactory on the
scales of interest, l&102 Mpc. Because perturbations do not
grow as fast as d r / r }a in a low density universe, we use the
factor g(V) @defined in Eq. ~5.5! # to modify the amplitude
of present-day perturbations @31#. Hence, we obtain the
power spectrum ~adapted from @9#!
4 p V 2 h 4 u 21 m 6 ~ V ! 2 k 4 g ~ V ! 2
model
u w ~ kR ! u 2 4 p k 2 P ~ k ! dk,
w ~ x ! 53 ~ sinx2xcosx ! /x 3 ,
~5.4!
56
F
G
2
1
,
111/~ u 5 k ! 2
~5.3!
which is observed to be around unity @34–36#. An excellent
fit to our results is given by the empirical formula
s 8 ~ V,h ! 50.25~ 60.1!
S
D
g ~ V ! G ~ 112.6G21.6G 2 !
3 m 6~ V !
,
V
11 ~ 10G ! 22
~5.5!
with
g ~ V ! [ 25 V/ @ 11 21 V1V 4/7# ,
which is valid to within ;10% for 0.1<V<1 and
0.4,h,0.8. The error bars on s 8 are estimated based on the
quoted uncertainties in the string parameters @24,25,29,30#
and the uncertainty in the CMB normalization of m @17#
included in G m (V), which we repeat are probably too small
for low V. Evaluating Eq. ~5.5! for various values of the
cosmological parameters, we predict s 8 (1.0,0.5)50.2560.1
and s 8 (0.2,0.5)50.0560.02. For V51 the string1HDM
scenario requires a modest boost or bias in the power in
order to achieve s 8 ;0.57–0.75 @34,36#. These results are in
agreement with past work by Colombi @37#, based on the
Bennett-Bouchet simulations. We pause to note that the nonlinear dynamics of wakes and filaments @9,38–49# may produce such a bias sufficient to reproduce the observed clustering of objects on large scales. However, in an open universe
the peak amplitude of k 3 P(k) drops and shifts to larger
scales, so that some sort of scale-dependent boost would be
required to produce more power for k*1 Vh 2 /Mpc. Hence,
string1HDM in an open universe does not appear to be a
viable model for structure formation.
Structure formation by strings with CDM in a flat, V51
universe, when normalized on large scales, suffers from producing too much power on small scales. As pointed out by
56
COSMIC STRINGS IN AN OPEN UNIVERSE: . . .
4575
@4–6# this problem may be overcome, as for standard CDM,
in a low density, V,1 universe. Examining Fig. 3, we see
that the string1CDM power spectrum ‘‘bends over’’ on
small scales as we lower V. Hence, for G[Vh;0.1–0.2,
the spectrum approximately fits the shape of the PD reconstruction. The variance of the mass fluctuation is given by
the empirical formula
We may estimate the CMB normalization of the mass per
unit length as a function of V m following the methods of
Sec. IV. However, there is no correction for the geometry,
since the spatial sections are flat. Hence, we find the empirical formula
g ~ V ! G ~ 120.36G !
,
V 11 ~ 50G ! 22
~5.6!
fits our results to within 5% for 0.1<V m <1. The subscript
L is used to differentiate the above normalization from the
case of an open universe, in Eq. ~4.6!. We see that the normalization is relatively insensitive to the presence of a cosmological constant.
Finally, we may consider the properties of the cosmic
string scenario for structure formation with CDM in the presence of a cosmological constant. We may adapt Eq. ~5.3! for
the string1CDM power spectrum by setting V5V m and using the appropriate growth factor @31#. The variance of the
mass excess on length scales R58h 21 Mpc is fit by the
empirical formula
S
s 8 ~ V,h ! 50.9~ 60.5! m 6 ~ V !
D
which is valid to within ;10% for 0.1<V<1 and
0.4,h,0.8.
Evaluating
Eq.
~5.6!,
we
predict
s 8 (0.4,0.7)50.460.2 and s 8 (0.2,0.5)50.260.1. Hence,
for G;0.1–0.2, the range of values of the mass fluctuation
excess
falls
well
below
the
estimate
of
2
s 8 50.6132%
224% exp@(20.3620.31V10.28V )lnV# @50# by a
factor of ;2 –4. Within the uncertainties quoted in Eq. ~5.6!,
a bias as low as b;1.5 may be needed. Recent work by
Sornborger et al. @44# on the structure of cosmic string
wakes has shown that the ratio of the baryon to CDM density
in wakes is enhanced. For a single wake formed near
radiation-matter equality, the baryon enhancement at late
times is ;2.4 in a region of thickness ;0.3 Mpc. These
results, which suggest that structure formation by strings is
biased, allow our conclusion that the string1CDM model
may be a viable candidate for the formation of large scale
structure in an open universe.
VI. COSMOLOGICAL CONSTANT
In this section we briefly consider the effect of a cosmological constant on the cosmic string scenario. The background cosmology in this case is a spatially flat, FRW spacetime with a cosmological fluid composed of vacuum and
matter components such that V m 1V L 51. The expansion
scale factor is given by the expression
a~ t !5
F
S
12V L
3
sinh2 H 0 t AV L
VL
2
DG
1/3
,
~6.1!
where H 0 , and V L are the present-day Hubble constant and
vacuum-matter density parameter. We may now follow a
similar procedure as outlined in Sec. III to study the evolution of the long string length scale L and velocity v by taking
the spatially flat, V→1 limit in Eqs. ~3.1! and ~3.2!. In this
case we find that for a comparable matter density as in an
open universe, the dilution of the string energy density and
the damping of string motion are much weaker in the cosmological constant universe. Note that the argument of the
sinh in the scale factor, evaluated at the present-day, is
1
2 lnu(11AV L )/(12 AV L ) u . Hence, for small V L the scale
factor behaves to leading order as a(t);t 2/3, just as for
matter-dominated expansion. Only when V L →1 are the effects of the exponential expansion important, damping the
string motion. For example, in the case of V m 50.3, the ratio
L/t is only ;5% larger and the velocity is only ;5%
smaller than the V m 51, spatially flat value. For the open
universe with V50.3, the ratio L/t has grown by ;15% and
the velocity has dropped by ;30% from their V51 values.
26 20.05
G m ~ V m ! L 51.0510.35
20.20310 V m
s 8 ~ V m ,h ! 50.9~ 60.5!
S
3 m 6~ V !
~6.2!
D
g ~ V m ,V L ! G ~ 120.36G !
,
Vm
11 ~ 50G ! 22
~6.3!
with
1
1
g ~ V m ,V L ! [ 25 V m / @ V 4/7
m 2V L 1 ~ 11 2 V m !~ 11 70 V L !# ,
which is valid to within ;10% for 0.1<V<1 and
0.4,h,0.8. We find that the amplitude of the string1CDM
power spectrum with a cosmological constant is higher than
in an open universe with the same matter density, as demonstrated in Fig. 3. Evaluating Eq. ~6.3!, we predict
s 8 (0.2,0.5)50.360.2 and s 8 (0.4,0.7)50.560.3. Comparing to observations, based on the estimate s 8
2
50.6 132%
224% exp@(20.5920.16V10.06V )lnV# @50# for a
spatially flat universe, we find that a slightly lower bias than
in an open universe, b;1.5–4, is required. Hence, the
string1LCDM scenario may be viable if the strings generate
a sufficient bias to explain the clustering on 8h 21 Mpc
scales.
VII. CONCLUSION
In this paper we have laid out many of the tools necessary
to study cosmic strings in an open universe. We have first
derived the equations of motion and energy conservation in
an V,1 FRW space-time. We have extended the MS model
of quantitative string evolution @12# to the case of an open,
V,1 universe. We believe this extrapolation is reasonable
for the range of values of V of interest. We have found that
with the onset of curvature dominated expansion, the long
string energy density and mean velocity decay rapidly. We
have shown that the resulting effect on the large angle CMB
temperature fluctuations induced by cosmic strings is a lower
level of anisotropy than in a critical, V51 universe, for the
same m . Constructing a semianalytic model for the generation of CMB anisotropy in an open universe, based in part on
4576
P. P. AVELINO, R. R. CALDWELL, AND C. J. A. P. MARTINS
56
the AS numerical simulation @29,30#, we found that comparison with the COBE-DMR observations @19# leads to a higher
normalization of the cosmic string mass per unit length. To
the extent that the CMB anisotropy induced by realistic cosmic strings has been accurately simulated in Ref. @17#, we
believe our results, Eqs. ~4.6! and ~6.2!, are reliable within
the errors discussed. The new normalization of m , the first
estimate of the normalization of m in a low density universe
~as far as we are aware!, is consistent with all other observational constraints on cosmic strings, including the bound on a
stochastic gravitational wave background arising from pulsar
timing @51#.
Finally, we have demonstrated the effect of an open,
V,1 universe on the power spectrum of density fluctuations
produced by cosmic strings with HDM and CDM. As we
mentioned in Sec. I, the power spectrum P(k) does not completely specify the cosmic string structure formation scenario. Fluctuations generated by string wakes and filaments
are non-Gaussian, so that knowledge of P(k) alone is insufficient to specify all the properties of the density field. Although the linear power spectrum ~5.3! is in agreement with
the results of Avelino @32# and Colombi @37# on a limited
range of scales, we are unable to make finely detailed comparisons with observations without more knowledge of the
distribution of cosmic string seeded density perturbations.
For example, it is not clear whether the estimates of the rms
linear fluctuation in the mass distribution @34,36# obtained
from the various galaxy redshift surveys, which depend
strongly on the Gaussianity of the initial density field, are
directly applicable to a theory with a non-Gaussian fluctuation spectrum. Nevertheless, we have found that the string
1CDM spectrum fits the shape of the PD reconstruction of
the linear power spectrum @34# for cosmological parameters
in the range G;0.1–0.2. We have computed the variance of
the mass fluctuation in a sphere of radius R58 h 21 Mpc,
requiring a bias b*2 for consistency with the inferred s 8 of
the linear density field. In the case of a cosmological constant, a slightly lower bias is required than for an open universe string1CDM spectrum with the same matter density.
These findings are similar to Ref. @6#, in which the product
bG m was estimated in order to fit the string1CDM spectrum
to the 1-in-6 Infrared Astronomy Satellite ~IRAS! QDOT
survey @52#, and to Ref. @5#, in which the effects of an open
universe on global defects, including global strings and textures, were considered. The results of Ref. @44# indicate that
the density of baryonic matter is enhanced in CDM wakes by
a factor of ;2.4, suggesting that a bias b;2 may be possible. It is clear that high resolution simulations, as Ref. @53#,
are necessary to further develop the cosmic string structure
formation scenario.
The results presented in this paper provide excellent motivation to continue investigation of the cosmic string scenario, which should be possible with the equations of motion
for strings and the normalization of m for V,1.
@1# A. Vilenkin, Phys. Rep. 121, 263 ~1985!.
@2# A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other
Topological Defects ~Cambridge University Press, Cambridge,
England, 1994!.
@3# M. Hindmarsh and T. W. B. Kibble, Rep. Prog. Phys. 58, 47
~1995!.
@4# David N. Spergel, Astrophys. J. Lett. 412, L5 ~1993!.
@5# Ue-Li Pen and David N. Spergel, Phys. Rev. D 51, 4099
~1995!.
@6# Pedro Ferreira, Phys. Rev. Lett. 74, 3522 ~1995!. Note the typo
in the definition of j which should read j [( r L / m ) 21/2.
@7# Xiachun Luo and David N. Schramm, Astrophys. J. 421, 393
~1994!.
@8# John L. Tonry, in Proceedings of the Texas/PASCOS 92 Meeting on Relativistic Astrophysics and Particle Cosmology @Ann.
~N.Y.! Acad. Sci. 688, 113 ~1993!#.
@9# Andreas Albrecht and Albert Stebbins, Phys. Rev. Lett. 68,
2121 ~1992!; 69, 2615 ~1992!. Note the typo in Eq. ~3!, where
a factor of 2 is missing from the numerator, and in the definition of j immediately following, which should read
j [( r L / m ) 21/2.
@10# D. Austin, E. J. Copeland, and T. W. B. Kibble, Phys. Rev. D
48, 5594 ~1993!.
@11# B. Allen and R. R. Caldwell, Phys. Rev. Lett. 65, 1705 ~1990!;
Phys. Rev. D 43, 2457 ~1991!; 43, 3173 ~1991!.
@12# C. J. A. P. Martins and E. P. S. Shellard, Phys. Rev. D 54,
2535 ~1996!.
@13# C. J. A. P. Martins, Phys. Rev. D 55, 5208 ~1997!.
@14# J. Garriga and M. Sakellariadou, Phys. Rev. D 48, 2502
~1993!.
@15# L. Perivolaropoulos, Phys. Lett. B 298, 305 ~1993!.
@16# N. Kaiser and A. Stebbins, Nature ~London! 310, 391 ~1984!.
@17# B. Allen, R. R. Caldwell, E. P. S. Shellard, A. Stebbins, and S.
Veeraraghavan, Phys. Rev. Lett. 77, 3061 ~1996!.
@18# A. Stebbins and R. R. Caldwell, Phys. Rev. D 52, 3248 ~1995!.
@19# A. Banday et al., Astrophys. J. 475, 393 ~1997!.
@20# E. L. Wright et al., Astrophys. J. 420, 1 ~1994!.
@21# B. Allen, R. R. Caldwell, S. Dodelson, L. Knox, E. P. S. Shellard, and A. Stebins, ‘‘CMB Anisotropy Induced by Cosmic
Strings on Angular Scales *159,’’ astro-ph/9704160, 1997.
@22# Ue-Li Pen, Uros Seljak, and Neil Turok, Phys. Rev. Lett. 79,
1611 ~1997!.
@23# Joao Magueijo, Andreas Albrecht, Pedro Ferreira, and David
Coulson, Phys. Rev. D 54, 3727 ~1996!.
@24# D. P. Bennett and F. R. Bouchet, Phys. Rev. D 41, 2408
~1990!.
@25# D. P. Bennett, in The Formation and Evolution of Cosmic
Strings, edited by Gary Gibbons, Stephen Hawking, and Tan-
ACKNOWLEDGMENTS
We would like to thank Chung-Pei Ma, Paul Shellard,
Andrew Sornborger, and Albert Stebbins for useful conversations. P.P.A. was funded by JNICT ~Portugal! under ‘‘Programa PRAXIS XXI’’ ~Grant No. PRAXIS XXI/BPD/9901/
96!. The work of R.R.C. was supported by the DOE at Penn
~Grant No. DOE-EY-76-C-02-3071!. C.M. was funded by
JNICT ~Portugal! under ‘‘Programa PRAXIS XXI’’ ~Grant
No. PRAXIS XXI/BD/3321/94!.
56
@26#
@27#
@28#
@29#
@30#
@31#
@32#
@33#
@34#
@35#
@36#
@37#
@38#
@39#
@40#
COSMIC STRINGS IN AN OPEN UNIVERSE: . . .
may Vachaspati ~Cambridge University Press, Cambridge, England, 1990!; F. R. Bouchet, ibid.
E. P. S. Shellard ~private communication!.
P. Mahonen, T. Hara, and S. J. Miyoshi, Astrophys. J. Lett.
454, L81 ~1995!.
Shoba Veeraraghavan and Albert Stebbins, Astrophys. J. 365,
37 ~1990!.
B. Allen and E. P. S. Shellard, Phys. Rev. Lett. 64, 685 ~1990!.
E. P. S. Shellard and B. Allen, in The Formation and Evolution
of Cosmic Strings @25#.
Sean M. Carroll, William H. Press, and Edwin L. Turner,
Annu. Rev. Astron. Astrophys. 30, 499 ~1992!.
Pedro Avelino, thesis, University of Cambridge, 1996.
Chung-Pei Ma, Astrophys. J. 471, 13 ~1996!.
J. A. Peacock and S. J. Dodds, Mon. Not. R. Astron. Soc. 267,
1020 ~1994!.
M. Davis and P. J. E. Peebles, Astrophys. J. 267, 465 ~1983!.
S. D. M. White, G. Efstathiou, and C. S. Frenk, Mon. Not. R.
Astron. Soc. 262, 1023 ~1993!.
Stephane Colombi, Ph.D. thesis, Universite de Paris 7, 1993.
Tanmay Vachaspati and Alex Vilenkin, Phys. Rev. Lett. 67,
1057 ~1991!.
Tanmay Vachaspati, Phys. Rev. D 45, 3487 ~1992!.
Anthony N. Aguirre, and Robert H. Brandenberger, Int. J.
Mod. Phys. D 4, 711 ~1995!.
4577
@41# P. P. Avelino and E. P. S. Shellard, Phys. Rev. D 51, 369
~1995!.
@42# R. Moessner and R. Brandenberger, Mon. Not. R. Astron. Soc.
280, 797 ~1996!.
@43# V. Zanchin, J. A. S. Lima, and R. Brandenberger, Phys. Rev.
D 54, 7129 ~1996!.
@44# A. Sornborger, R. Brandenberger, B. Fryxell, and K. Olson,
Astrophys. J. 482, 22 ~1997!.
@45# Dan Vollick, Astrophys. J. 397, 14 ~1992!.
@46# Dan Vollick, Phys. Rev. D 45, 1884 ~1992!.
@47# Tetsuya Hara, Hideki Yamamoto, Petri Mahonen, and Shigeru
J. Miyoshi, Astrophys. J. 432, 31 ~1994!.
@48# Tetsuya Hara, Hideki Yamamoto, Petri Mahonen, and Shigeru
J. Miyoshi, Astrophys. J. 461, 1 ~1996!.
@49# Petri Mahonen, Astrophys. J. Lett. 459, L45 ~1996!.
@50# Pedro T. P. Viana and Andrew R. Liddle, Mon. Not. R. Astron. Soc. 281, 323 ~1996!.
@51# R. R. Caldwell, R. A. Battye, and E. P. S. Shellard, Phys. Rev.
D 54, 7146 ~1996!.
@52# Hume A. Feldman, Nick Kaiser, and John A. Peacock, Astrophys. J. 426, 23 ~1994!.
@53# A. Sornborger, Phys. Rev. D ~to be published!.